Milnor invariants of braids and welded braids up to homotopy
Résumé
We consider the group of pure welded braids (also known as loop braids) up to (link-)homotopy. The pure welded braid group classically identifies, via the Artin action, with the group of basis -conjugating automorphisms of the free group, also known as the McCool group P dagger n . It has been shown recently that its quotient by the homotopy relation identifies with the group hP dagger n of basis -conjugating automorphisms of the reduced free group. We describe a decomposition of this quotient as an iterated semidirect product which allows us to solve the Andreadakis problem for this group, and to give a presentation by generators and relations. The Andreadakis equality can be understood, in this context, as a statement about Milnor invariants; a discussion of this question for classical braids up to homotopy is also included.